diagonal vs regular

[Gottfried]

So, in the finite case, for P~ * Bb * P^(-1)~ = X we don't get a triangular X, although it will be exactly "similar" to Bb(truncated), in the sense, that the eigenvalues, eigenvectors etc obey all known rules for similarity-transform.

Sorry, I dont get the point of this. What are "similarity conditions", what are "all known rules for similarity-transform"?

"Similarity transform(ation)" in the sense of linear algebra.

If X = A * B * A^-1 then X is said to be "similar" to A; this means, it has for instance the same eigenvalues. Also this "similarity transform" is transparent for some matrix-functions like powers, exp(), log() etc.

log(X) = A * log(B) * A^-1 .

The special case is, if X is diagonal (by appropriate selection of A), then X contains the eigenvalues of B in its diagonal. And so on.

In the case of infinite dimension we may have, that the inverse is not unique, also we call it the "reciprocal" instead. Say Z defined to be a reciprocal to A, so that
A*Z = I
for the case of infinite size, then we may have different Z with the same reciprocity-relation.
A*Z1 = A*Z2 = A*Z3 =...= I

Also, - what I have learned here - we may have different A for a given B, such that not only

X1 = A1 * B * A1^-1

but also
X2 = A2 * B * A2^-1
...
with X1<>X2<>... and all Xk being diagonal

Then apparently it follows also, that we have multiple diagonalizations resulting in different X1,X2,X3,...
X1 = A1 * B * Z1_1 = A1 * B * Z1_2 = ...
X2 = A2 * B * Z2_1 = A2 * B * Z2_2 = ...
...
(However, I'm asserting the latter just in this notepad-entry the first time, so maybe this is a bit of overgeneralization)
[update 1.5.08] see also example
where I already computed one example for this [/update]

Yes, for the cases of b in the range of convergence.

What is the "range of convergence"?

ehmm... 1/e^e < b < e^(1/e), sorry.

The eigenvalues v_k (k=0..inf) are then |v_k| <= 1 and also v_k=(v_1)^k

Hmm. Because of the case of multiplicity of possible sets of eigenvalues we should perhaps introduce the convention to denote one set as "principal" set, for instance when we use the similarity-transformation to implement the shift to the attracting fixpoint.